2015 Tenth International Conference on Ecological Vehicles and Renewable Energies (EVER)

Simplified model of eddy current brakes and its use for optimization

Christoph Holtmann Institute of

Frank Rinderknecht Institute of

Horst E. Friedrich Institute of

Vehicle Concepts

German Aerospace Center

70569 Stuttgart, Germany

Email: christoph.holtmann@dlr.de

Vehicle Concepts

German Aerospace Center

70569 Stuttgart, Germany

Email: frank.rinderknecht@dlr.de

Vehicle Concepts

German Aerospace Center

70569 Stuttgart, Germany

Email: horst.friedrich@dlr.de

Abstract-The electrification of vehicles makes it possible to think about new concepts to substitute conventional technologies. For example in an electrical drive train, with a traction motor for each wheel mechanical friction brakes can be replaced by eddy current brakes. In this work a method for optimizing the torque density of eddy current brakes is presented. For a fast optimization process the model of the eddy current brake is split up in two levels of detail. In the first level a global model including a simplified eddy current model and a lumped parameter magnetic circuit to calculate the primary magnetic flux will be described. The second level includes a detailed eddy

current model and will be described in further works. The simplified eddy current model include fitting parameters which allow to fit the torque curves to the ones of the detailed eddy current model. As a result, the method allows to optimize the entire system represented by the global model in a very fast way. The development of the global model, its verification and the method for optimizing the entire system represented by the global model is shown.

I. INTRODUCTION

The most important parameters of eddy current brakes are their torque density and their torque characteristics. The analysis of the torque characteristics is only possible with simplified analytical models or FEM models. In [4] the torque characteristics of an eddy current brake with different copper layer thicknesses were calculated by using FEM analysis. The results were fitted with a characteristic function to analyse the braking performance during vehicle-dynamic simulations. In this work a model based fitting function is used to analyse the braking performance instead of a parameter-based fitting function (see equation (1) in [4 D. The model based fitting function is a simple impedance model as in [l]. For a fast optimization process the analysis of the eddy current brake is split up in two models.

The first one, global model, describes the behaviour of the eddy current brake. The second one (detailed model) models the behaviour of the eddy currents in the eddy current area due to the air gap and the groove slot. The detailed model will be discussed in future works. To analyse the torque density of the entire eddy current brake the mass and volume of the magnetic excitation circuit has to be considered. Therefore the global model consists of a lumped parameter magnetic circuit and a simplified eddy current model. The connection between the global model and the detailed model is done by fitting the

978-1-4673-6785-1/15/$31.0002015 European Union

torque curves of the detailed model with the simplified eddy current model. In this work the development of the global model is described and its verification and the use of the optimization process is shown.

A. Characteristic quantities The goal of the optimization is to find the geometric

parameters of the eddy current brake (ECB) which result in a maximum torque density. Unlike electrical machines; which have a constant torque up to the nominal speed, the torque of ECBs increases up to a critical speed where the torque is at its maximum. At speeds higher than the critical speed the torque decreases. Figure I.1 shows qualitatively the torque curve of an ECB with its characteristic values. It can be seen that the torque density of ECBs can be defined in many ways. In past

M ft1max+-------��-- requked _c!Jl}�� �o!,g':!f! _

Fig. 1.1: characteristic torque curve of an ECB

n

works the peak torque lvlmax and the critical speed ncrit were analysed. With the volume of an ECB Vecb the peak torque density is

G _ lvlmax max - Vecb

. (1)

For many applications the peak torque is not the most im­portant factor. When a constant torque is required over the operation range, as illustrated in figure I.l, the torque density should be defined as the product of the middle torque density

and the ratio of the middle torque density and the peak torque density. The middle torque density

1 in=nmo" Gm = M(n)dn

nmax Veeb n=O (2)

could be identical for a flat torque curve and a torque curve with a high peak torque. When a constant torque is required and the torque curve has a high peak torque, the torque potential is not fully used. To design an ECB with a high torque in a wide operation range, the middle weighted torque density Gmw has to be maximized. Gmw is defined as

Gmw = 1 1 [ I

n·n==onmo" l'vl(n)dn]

2 (3) nmax Vecb lvImax iT

B. Eddy current model The eddy current model used in this work consists of a

conducting plate moving with the velocity Vm through an inhomogeneous primary magnetic field, represented with the magnitude 6B = B+ - B- of the air gap flux density. The eddy currents are approximated in a single eddy current path with the resistances R1, R2 and R3 as shown in I.2(a).

vm� -

Sm Sea

Seae eddy current path

j

t(y)

y (a) (b)

Fig. 1.2: Model of a conducting plate moving with the velocity Vm through an inhomogeneous primary magnetic field while approximating the eddy currents in one eddy current path (a). Model for the effective thickness of the eddy current path (b).

During the movement with the middle velocity Vm, the induced voltage Ui in the eddy current path will be calculated by using Faraday's law of induction.

is the formula for a voltage induction path length of 0.5b. This length is also used to calculate the resistances Rl and R3 of the eddy current path.

(5)

(6)

The total resistance of one eddy current path is

R = 2CR(R1 + R2)' (7)

The resistance factor CR is used in equation (7) because the eddy currents are approximated only in one eddy current stream. The resistance factor is one of the fitting parameters to connect the detailed model with the simplified eddy current model.

To approximate the field repression the resistances are calculated with the effective thickness of the eddy current cross section 8coe. 8eeo is the thickness related to a fictive current 1fiet in the homogeneous eddy current path. This fictive current is approximated by integrating the current density in a conducting half space on which a pulsating magnetic field is applied. The boundaries of the integration are the surfaces y = 0 and y = 8eo as shown in figure 9 (b).

1fict = i(y)dy = ifict8COe i sno

with y

i(y) = ise5

. y=O

(8) This results in the effective thickness of the eddy current area

(9)

where 6 is the penetration depth with the relative permeability

ILrco and the conductivity IJ" of the eddy current material.

T (10)

JrVmlLOILrcolJ"

In this context the relative permeability fLreo is just a fitting parameter because the eddy current material has a non linear behaviour and is inhomogeneous in space.

As in equation 1 in [1] an inductance L is applied into the eddy current path, because the eddy currents are approximated only in one current stream. It is assumed that the inductance first of all is a result of the magnetic resistance of the air gap.

L - _1 _ _ fLOCLTb 11) -

Rmi5 -

88 (

Because the eddy currents are inhomogeneous the inductance factor CL is used in equation 11 as a fitting parameter. The inductance results in an reactance

X = Lw = L2Jrvm

. (12) T

Due to the impedance 2!.. = R + iX and the induced voltage, the current in the eddy current path is

I = � = Ui

12!..1 vi R2 + X 2 (13

)

Equation (5) in [1] is quite similar to the equation above but the resistances are calculated differently. The model of [1] does not include the effect of the field repression represented by the effective thickness of the eddy current path as described in equation (9).

Due to the total power dissipation of the ECB P 12 RNeep and the conservation of energy, the drag force of an eddy current brake can be approximated with

P 6B2b2v F =

Vm =

4(R2 + X":;) RNeep, (14)

where Necp is the number of eddy current paths.

The eddy current model is verified by using the results from FEM analysis and measurements of other publications [2] [3] [4]. The model should be independent of the magnitude of the ECB therefore papers with different sizes of eddy current brakes were chosen for the verification. The smallest was the ECB investigated in [2] and the biggest was investigated in [4]. To verify the ECB model equation (14) the torque or force curves from [2], [3] and [4] are fitted by using the least square method with the resistance factor CR, the inductance factor CL

and the relative permeability /Lr as fitting parameters. Because the primary air gap flux density 6B is not explicitly given in [2], [3] and [4] the magnitude of the air gap flux density is also used as a fitting parameter. The torque curves from [2], [3] and [4] are transformed to force speed curves with the middle active radius r musing

F = l'vljrm and Vm = rmW = rm27rn. (15)

This allows the comparison of the radial-f1ux-ECB in [3] with the axial-f1ux-ECBs in [2] and [4]. The geometric data from [3] is extracted from figure I and 2 in patent [9] while the rotor diameter is given in figure 3 in [9]. From [3] the torque curve given in [9] with no copper layer is used. It must be pointed out that the velocity in figure I in [4] is the train speed. The radius of the wheel is O.460m while the middle active radius is rm = O.2625nL The parameters utilized for the verification from [2], [3] [9] and [4] are shown in table I.1.

Figure I.3 shows qualitatively the result of the verification. Due to the fitted torque curves, the quantitative results are

1

0.8

Bo Zhang

.�GOGOQ eP .§:t.

r.\ .... + '!-' S.Anwar¥--i"" o /-f 6 ?I-

0.6 0 71-S.E.Gay

CD -f . / 0.4 : /'

Q+ 0.2 . /

o�--�----�----�----�--� o 0.2 0.4 0.6 0.8 1

Fig. 1.3: Braking force vs. speed of different ECB, data from [3, S. Anwar],[4, Bo Zhang] and [2, S.E.Gay] (markers) and model based fitted curves (lines) Vm.ax = 9.68m/ s Fm.ax = 44.47 N (S.E.Gay), Vm.ax = 22.26m/ s, Fm.ax = 6145N (S.Anwar), Vm.ax = 55.42m/ s, Fm.ax = 16086N (Bo Zhang).

shown in table I.2. The maximum error related to the maximum torque

W!it - Fre! I e - -'------''--------''--'-

� -

Fmax,re! ( 16)

from every fitted torque curve is less then 4 %.

TABLE 1.1: Parameter of the ECBs from [2], [3] and [4] needed for the verification, the conductivity of the ECB from [3] \[9] is not explicitly given, a conductivity of carbon steel is used(l)

Parameter Bez. Gay [2] S. Anwar [3] Bo Zhang [4] middel active rm[m] 0.0508 0.1225 0.2625 radius

width eddy

current- b[m] 0.0254 0.095 0.232 element

thickness of eddy current sGo [m] 0.00317 0.0086 0.0325 element

width of one pole T[m] 0.04 0.064 0.412 pair

number of eddy current N"p[l] 16 24 8 pathes

conductivity reference o-[O-"m-"] 17e6 4.5e6(1) 4.55e6 [2] [3] [4]

TABLE 1.2: Fitted parameter of the ECBs from [3], [2] and [4]

Parameter Bez. Gay [2] S. Anwar [3] Bo Zhang [4] amplitude air gap bE fit [T] 0.935 2.48 1.62 flux density

resistance 0.857 1.1 2.02 factor CR,

inductance 0.965 0.2 0.21 factor CL

relative 7.5 29.2 8.2 permeability (fit) /-LT' f �i, r

maximum 0.022 0.032 0.023 error

eTno:r

To evaluate experimental results only the resistance factor CR, the inductance factor CL and the relative permeability /Lr will be used because the primary magnetic field can be measured. The primary magnetic field is not explicitly given in the publications of [2], [3] and [4]. In [4] the torque characteristics for different air gap thicknesses was analysed. In this work a 2-D-FEM model of the ECB from [4] is built up to analyse the air-gap flux density. The maximum flux density is used to fit the curves given in figure 10 in [4] while using CR,

TABLE 1.3: Fitted parameter of the ECB from [4] with the variation of the air gap thickness.

8[mm] 3 5 8 12

bE[T] 1.68 1.62 1.54 1.46 CR, 1.99 2.02 2.02 2.576 CL 0.168 0.194 0.223 0.242

/-L·r j' ,it 5.45 8.2 11.58 7.6 eUtn:r 0.023 0.03 0.033 0.023

CL and ILr as fitting parameters. Figure I.9 shows the results of the 2-D-FEM calculation of the air gap flux density on the rotor surface. The maximum flux density due to the different air gap thicknesses and the fitting results of CR and CL are given in table I.3.

It can be seen that the correlation between the fitting parameters (resistance factor, the inductance factor and the relative permeability) and the air gap, cannot be described in a simple way. This correlation will be more precisely analysed with the detailed model in future papers.

To test the optimization process of the global model developed in this work, the resistance factor CR, the inductance factor CL and the relative permeability /Lr are set to one.

C. Dimensionless parameters and geometry The optimization should be independent of the ECB's

magnitude and the analysis of the torque density is done with dimensionless variables and parameters. The pattern of eddy currents and therefore the torque characteristics first of all depends on the magnetic Reynolds Number as shown in [5]. The middle velocity Vm of the moving eddy current material, the width of one pole pair T and the conductivity are the characteristic quantities.

(17)

For the eddy current brake optimized in this work, as shown in figure 1.4, the velocity is Vm = Ta(1 +Ti)W /2 and the width of one pole pair is T = Ta7T(1 + Ti)/Npp. This results in the magnetic Reynold Number

T; * 2 RmT = -N

(1 + Ti ) WIJ'/Lo 2 pp

with the outer radius T a as characteristic quantity.

( 18)

One of the most limiting parameter of ECBs is the current density of the excitation coils icmax = jo. In the case of a linear magnetisation curve the primary magnetic field is proportional to the excitation current Icmax rv icmaxT; due to the magnetic resistance of the magnetic excitation circuit Rmep rv l/(/LoTa). This results in the reference flux density

Eo = icmaxlLOT a' Magnetic resistances are related to Rmo = 1/ (/LOTO)

R";" = Rm

= Rm /LoTo Rmo 5*

( 19)

(20)

and magnetic strain to Vo related to Ro = l/(IJ'Ta).

T5io. Electrical resistances are

The geometry of the ECB, which is optimized in this work, is shown in figure 1.4. For the optimization process geometric parameters are defined which can only reach values from zero to one. All geometric parameters are directly or indirectly related to the outer radius T a and are always related to its possible maximum except of the aspect ratio A and the air gap 5*. The definitions of the geometric parameters are given in table 1.4.

8-8

V;n ----

Fig. 1.4: geometry of the ECB optimized in this work

TABLE 1.4: Definition of geometric parameters

ratio

aspect ratio

inner radius to outer radius

air gap to outer radius

width rotor to max. width Rotor

width eddy current element to max. width eddy current element

width groove to max. width groove

hight groove with groove slot to width rotor

width groove slot to width groove

higth groove slot to high! groove with groove slot

def.

8* = � TO

b.*. = __ b_.,. _ = br , ro2.\-8 ro(2.\-8*)

S* _ Seo Seo GO - ro2'\ -8 -br ro(2'\ -8*)(1 -b;)

h* = hq8 = hq8 g.; hq roh�b; (2,\ -8*)

The application of the dimensionless parameters to the equations of the eddy current model gives for equation (7)

where 8�oe (see equation 9) is

(22)

with 1f(1 + rn

N£pRmT fJrco (23)

Equation (12) using equation (11) results in

X* -CL Npp(1 - rn R - 6* 4(1 + rn mT' (24)

The induced voltage can be written in dimensionless form as

* A B* Npp(l - rn R (25) ui = L',

0 21f(1 + rn mT'

With P = U;R/Z2 and M = P/w the torque in dimensionless form is

(26)

The dimensionless form of the torque density is

6.B*2(1 - r*)2 N2 G* = 0

, t pp R* RmT (27)

8A 1f2(X*2 + R*2)

D. Lumped parameter magnetic circuit To approximate the air gap flux density Bo a lumped

parameter magnetic circuit shown in figure I.5 is modelled. The current in the excitation coils causes the magnetic strain Vm. In dimensionless form the magnetic strain is

V';; = 2b;r; sin (_1f_) h;b;(2A - 6s)(1 - h;s)' (28) 2Npp

The magnetic resistances Rm2 and Rm3 are calculated with the respective length of the flux path Ii, its cross section Ai and its permeability fJi = fJOfJri

R _ Ii , - fJOfJriAi

The magnetic resistances Rm1, Rm4 and Rm5 are the sum of the magnetic resistances of flux paths in radial and axial direction. Given from [7] the magnetic resistances Rmo and Rm6 are

1 Rm1 = Rm6 = with t = ro(1 - r;) (29) fJoO.26t

The magnetic resistances are given in dimensionless form in table I.7. The linear equation system including 5 mesh and 4 node equations is solved iteratively using the magnetisation curve of steel 1008 for the resistances Rm1, Rm2 and Rm5. The relative permeability is interpolated as a function of the flux density while using the flux density in the respective cross section. For interpolating the real flux density

B = BoB* = jcmaxfJor oB* (30)

is needed. Hence the optimal design parameters in table 1.4 depend on the main dimension, the outer radius r o'

Fig. 1.5: Lumped parameter magnetic circuit

The lumped parameter magnetic circuit is verified with a 2D-FEM-analysis using the parameters in table I.5 with b; = 0.7 and b; = 0.8.

TABLE 1.5: Parameters that are used to verify the lumped parameter magnetic circuit with a 2-D-FEM-calculation

ro[m] A 0* [1] NeeI'[l] r; [1] 0.15 0.2 0.01 4 0.5 s�o [1] b; [1] b;, [1] h;[l] h;, [1] 0.5 0.5 0.6 0.2

Figure I.6 shows the flux density magnitude in areas with the highest saturation. The discrepancy between the analytical and FEM solution in this areas is less then 10%.

To examine the connection between the eddy current model and the lumped parameter magnetic circuit the torque curves for different excitation currents are calculated using the geo­metric parameters given in table I.6. The calculation is done by using the simplified eddy current model. This includes the air gap flux density that is calculated with the lumped parameter magnetic circuit. The resulting torque curves can be compared with the torque curves in figure 8 in [4], because the same geometric parameters are used. For the calculation the fitted parameters for an air gap of 5mm and an excitation current of 88A are used (CR = 2.02, CL = 0.194 and fJr = 8.2, table 1.3).

Figure I.7 shows the torque curves calculated with the simplified eddy current model and the lumped parameter magnetic circuit (lines) and the torque curves as the result of a FEM-calculation of [4] for different excitation currents (I = 52A; 76A; 100A; 124A). It can be seen that the torque curves calculated with the global model for a excitation current of I = 52A, I = 100A and I = 124A do not fit very well to the curves of [4]. This is because the relative permeability in the eddy current area changes due to the not linear magnetisation curve.

1,37

1,34

1,73 T analy.

(a)

1,3 T (analy.)

(b)

1,58 T analy.

1,88 T (analy.)

25 r--�--�--�--�--�----'

20

15

<J <J <J <J <J <J <J <J

<J ++ + +++ ++ +++ <J+ . ++++ +./ . - . ... +

<J 07'V'V"V'V 9'V.s;J � . - • � ........ . +/V 'V-V�'l-';;- . - .

10 <J t;

V""9 9V

5

<J' x x x x x x x xx xx x xx x

o.---�--�----�--�--�--� o 10 20 30

vm[m/s] 40 50 60

Fig. I.7: Braking force vs. speed for ditlerent excitation currents, data from [4] (markers) and torque curves based on the global model (lines) I = 52A crosses and solid line, I = 76A down triangles and dashed line, I = lOOA + and dash dot line and I = 124A left triangles and doted line.

Fig. 1.6: Magnitude of the flux density in areas of highest saturation of a 2D-FEM analyses (a) b; = 0.7 (b) b; = 0.8 using the parameters in table 1.5 TABLE 1.7: dimensionless magnetic resistances

TABLE 1.6: Parameters, extracted from [4], that are used to examine the connection between the simplified eddy current model and the lumped parameter magnetic circuit

ro[m] NecI'[I] TI'I'[m] sco[m] O"[O-'m-'] ,,\[1] 0* [1] 0.375 8 0.412 0.0325 4.55e6 0.197 0.013 r; [1] b;. [1] <0[1] b; [1] b;, [1] h;[I] h;, [1] 0.4 0.745 0.7 0.18 0.64 0.21

E. Optimization method of the global model An interval search method as described in [8] is used

to determine the geometric parameters which result in a maximum torque density. The unknown function is calculated in an interval Xleft Xright with two inner points as shown in figure L 1 0, If one inner point is the maximum value of the four points, the neighbour points of this point will be chosen as the next interval bounds. If the maximum value of the four points is located at the right bound, the next interval is between the point left to the maximum point and the point one step right of the maximum point. The same, in a mirrored way, will be done if the maximum value of the four points is located at the left bound, The process is repeated until the difference of one location where the maximum occurs Xmax (i) and the location of the maximum one step before Xmax (i - 1) is

6x = IxmaxU) - xmax(i - 1)1 < 6xmax·

This method works because the maximum of the torque density is a global maximum for each parameter. This method is used to optimize each parameter while using a multidimensional

region equation

air at rotor

rotor

groove slot

air gap

eddy current element

stator back

�(1 + rn + b�r; sin (�) . 1 2NI'I' . 2NI'I' R;nl = -j.L-rl-' b�(2"\ -0*)(1 - h�)(1 -rn 1 b.;.(2"\ -0*)(I+h;) + -- ------------"----;--.,,-

j.L,12 �(1- sin (�) Npp Npp

0* R;n3 = ---------------,-----,--

_1r (1 _ sin (_7r ) 2�p 2�p

1 [ SZ)0(2,,\-0*)(I-b;) R� ,4 = -- --------"'-"-'----------'--'-------'-'.-. --'--( -7r--'--) 1',·4 _7r_(I_ sm __ 2�p 2�p �(1

. + r:) + b;,b;r; sin (_

.7r ) 1 2NI'p 2NI'p + S* (2,,\ -0*)(1 -b*)(1 -r*) C(J 7" t

R* __ 1_ (2,,\ -0*)(1-S;;o(l-b.;) -b.;) m5 - j.L,.51 _1r_(1 _ _ (r* _ r*2)b ,b* sin (_1r_) 2Npp 't 't g.. y 2Npp

�(1 + rn + sin (�) + 1 2Npp Npp

roj.Lol',52 (2,,\ -0»(1 -S;)o(1-b�) -b�)(1 -rn +_1_ S;;o(2,,\ -0*)(1 -b.;)

j.L,4 _1r_(1 _ sin (_7r_) 2�p 2Nw

I' T

coils A

stator 7 """

� �� rotor

Fig. 1.8: 2-D-FEM-Model to calculate the air gap flux density from the ECB of [4]

1 0.75

0.5 0.25

h ;: 0

"0 CQ -0.25 -0.5

-0.75 -1

0 0.2

· · · · · · · 3mm . - . _ . 5mm

-Smm --- 12rmn

0.4 0.6 'PIT

O.S 1

Fig. 1.9: Calculated air gap flux density at the rotor surface at different air gap thicknesses using the geometry from the ECB of [4]

optimization method. The algorithm for optimising N param­eters contains N nested loops as shown in figure 1.11. The inner loop searches for the optimal parameter Xl that results in a maximum torque density by calculating the global model while using the parameters that variate during the outer loops. Each parameter will be optimized using the optimal parameters that are found in the inner loops, except of the first. After the first optimal parameter in an inner loop is found during the parameter variation of a loop, this parameter is used as the start value of the interval search method of the inner loop for the next step of the loop. This decreases the convergence time of each inner loop during the interval search method.

The optimization of the dimensionless middle weighted torque density G;;"w is done by using the constant parameters in table 1.8,a maximum magnetic Reynolds number, corre­sponding with nmax, of Rmmax = 1 and a convergence criterion of 6xmax = 0.02. The width b�s and the hight h�s of the groove slot will be optimized with the detailed model. The optimization of five parameters took three hours using Matlab R2007b on a Intel(R) Core(TM)i7-3770 @ 3,4 GHz.

Figure 1.12 shows the verification of the optimization method. It is done by the variation of each parameter while the others are at their optimum. It can be seen that the optimal geometric parameters are found with the optimization method

TABLE 1.8: Constant parameters of the optimization process

ro[m] Npp[l] jq[Am-2] O"[n-'m-'] 8* b�J1] h�J1] 0.150 8 6e6 5e6 0.005 0.5 0.1

because all parameters result in the same torque density at their maximum represented with the horizontal dashed line.

max 1 step 1

right bound step 1

right bound step 2 step 2

parameter Xj Fig. 1.10: interval search method

.--

� ,----

� ,----

global model

opti. para. 1

! x10 opti.

para. 2 using opt.

para. 1 �Xl0 �X20

opti. para. N

using opt. para. 1,2, ... N-1

--

-

---

X�o -

---

X�o

optimal parameters

Xl X2

X�o

Fig. 1.11: global optimization algorithm

II. CONCLUSION

In this work the development of a simplified eddy current model is shown. The connection between the detailed model and the simplified eddy current model including fitting param­eters is done by fitting the torque curves of ECBs analysed in other works with the simplified eddy current model. The

1.5 A

1.25

1 ;oJ

* 1: 0.75 \j 0.5

0.25

0.2 0.4 0.6 geo.par-ameter-

b* T

0.8

b* 9

1

Fig. 1.12: verification of the optimization method, the parameter for a respective line are signed up over the respective maximum

error of the fitting is less than 4 %. The relation between the geometries of the eddy current area and the pole tooth with the fitting parameters has to be analysed more precisely with experiments and the detailed model. This will be presented in future publications.

The lumped parameter magnetic circuit is verified with an error of less than 10% with a 2-D-FEM-model. Because the flux density is inserted with a square to the torque equation, the lumped parameter magnetic circuit has to be improved. Using the torque of different excitation currents, the connection between the lumped parameter magnetic circuit and the simpli­fied eddy current model is examined by using the results of [4]. Because the material behaviour is not linear, the connection between the simplified eddy current model and the lumped parameter magnetic circuit has to be analysed by using the fitting parameters as a function of the primary magnetic flux. This will be done with the detailed model and experimental measurements. It is shown that the optimization method of the global model works with a accuracy of 2%. It has to be pointed out that the accuracy of the optimization of a real eddy current brake depends on the accuracy of the global model. The accuracy, as mentioned before, will be improved with the detailed eddy current model. Up to now there is no indication that the optimization method will work with a higher convergence time when the fitting parameters are more precisely analysed with the detailed model.

ACKNOWLEDGMENT

The authors would like to thank all colleagues and students working at the eddy current brake project.

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